5 problems

Compact biconservative-surface conjecture. Any compact biconservative surface in $N^3(c)$ is $CMC$, that is, it has constant mean curvature.

Classification conjecture. Up to isometries of the domain and codomain, $M$ and $\varphi$ are those given in Theorem [the theorem identifies the explicit model family in the source…

A regular surface in $\mathbb{R}^3$ is a surface defined by an embedding. Let $\tilde{S}_{\tilde{C}_0}$, with $\tilde{C}_0>0$, denote the reference family of complete non-constant-…

Uniqueness conjecture. The only simply connected complete non-constant-mean-curvature biconservative surfaces in $\mathbb{R}^3$ are those given by that theorem. This is proposed as…

Let $M^2$ be a biconservative surface in $\mathbb{R}^3$, with mean-curvature function $f$, and let $U$ be a non-empty open subset of $M$. Uniqueness conjecture. If … on $U$, then ……